## Leonard Scott (University of Virginia)
## Friday 3rd November, 12.05-12.55pm, Carslaw 373## Reduced standard modules and 1-cohomology(This is joint work with Ed Cline and Brian Parshall.) First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles's famous paper on modularity of elliptic curves. Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott.
One can easily pass to
the case where the group acts faithfully, and the underlying
module is absolutely irreducible. In this case, R. Guralnick
conjectured that there is a universal constant bounding
all of the dimensions of these cohomology groups.
The work
described in this talk provides the first general positive results
on this conjecture, proving that the generic 1-cohomology
H
This result emerges as a
consequence of a general study, of interest in its own right, of
the homological properties of certain rational modules
Delta(lambda), Nabla(lambda), indexed by dominant weights
lambda, for a reductive group G. The modules
Delta(lambda) and Nabla(lambda) arise naturally from
irreducible representations of the quantum enveloping algebra
U |